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Theorem sbalex 2244
Description: Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb 2075.

That both sides of the biconditional express proper substitution is proved by sb5 2276 and sb6 2095. The implication "to the left" is equs4v 2011 and does not require ax-10 2145 nor ax-12 2179. It also holds without disjoint variable condition if we allow more axioms (see equs4 2417). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2461 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2460 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2269 in place of equsex 2419 in order to remove dependency on ax-13 2373. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2075. (Revised by BJ, 21-Sep-2024.)

Assertion
Ref Expression
sbalex (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sbalex
StepHypRef Expression
1 nfa1 2156 . . 3 𝑥𝑥(𝑥 = 𝑡𝜑)
2 ax12v2 2181 . . . 4 (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
32imp 410 . . 3 ((𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
41, 3exlimi 2219 . 2 (∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
5 equs4v 2011 . 2 (∀𝑥(𝑥 = 𝑡𝜑) → ∃𝑥(𝑥 = 𝑡𝜑))
64, 5impbii 212 1 (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1540  wex 1786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-10 2145  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-ex 1787  df-nf 1791
This theorem is referenced by:  equsexv  2269  sb5  2276  dfsb7  2284  mopick  2629  alexeqg  3550  dfdif3  4015  pm13.196a  41610
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